Question: Simplify
\[\frac{1 + \sin x - \cos x}{1 + \sin x + \cos x}.\]The answer will be a trigonometric function of some simple function of $x,$ like "$\cos 2x$" or "$\sin (x^3)$".
Explanation: From the double angle formulas, $\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}$ and $\cos x = 2 \cos^2 \frac{x}{2} - 1 = 1 - 2 \sin^2 \frac{x}{2},$ so
\begin{align*}
\frac{1 + \sin x - \cos x}{1 + \sin x + \cos x} &= \frac{1 + 2 \sin \frac{x}{2} \cos \frac{x}{2} - 1 + 2 \sin^2 \frac{x}{2}}{1 + 2 \sin \frac{x}{2} \cos \frac{x}{2} + 2 \cos^2 \frac{x}{2} - 1} \\
&= \frac{2 \sin \frac{x}{2} \cos \frac{x}{2} + 2 \sin^2 \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2} + 2 \cos^2 \frac{x}{2}} \\
&= \frac{2 \sin \frac{x}{2} (\cos \frac{x}{2} + \sin \frac{x}{2})}{2 \cos \frac{x}{2} (\sin \frac{x}{2} + \cos \frac{x}{2})} \\
&= \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}} \\
&= \boxed{\tan \frac{x}{2}}.
\end{align*}